All Questions
64 questions
2
votes
1
answer
161
views
Closed form expression for this zeta-like series involving GCD and LCM
I am looking for a closed form for this function $\Lambda:\mathbb{Q}^+\to\mathbb{R}^+$:
$$\Lambda(q) = \sum_{m,n\geq 1}\left(\frac{q\wedge\frac{m}{n}}{q\vee\frac{m}{n}}\right)^\alpha\left(\frac{m \...
- 634
2
votes
1
answer
202
views
Exponential sums involving smooth truncated divisor functions
Let $p$ be a prime, $a \neq 0$ an integer, let $M,N \gg 1$ and let $\psi,\eta$ be some fixed Schwartz functions. Would you know of any references in the literature where upper bounds for sums such as
$...
- 253
6
votes
2
answers
684
views
Number of divisors which are at most $n$
I’m interested in the function $\tau_n:\mathbb{N}\to\{1,2,3,\cdots, n\}$ defined by
$$\tau_n(x)=\sum_{k=1}^n \mathbf{1}_{k\mid x},$$
the number of divisors of $x$ which are at most $n$. Question 6 of ...
- 541
2
votes
0
answers
156
views
Exponential sum of $k$-fold divisor function
Can anyone point me to a reference for the main term when approximating the exponential sum of the 3-fold divisor function? Specifically I want the main term in $$\sum _{n\leq x}d_3(n)e\left (an/q\...
- 1,381
7
votes
2
answers
997
views
Convolution sum of divisor functions
Let $\sigma_0(n)$ be the divisor counting function
$$\sigma_0(n) = \sum_{d \vert n} 1.$$
I'm interested in the convolution sum
$$ S(n) := \sum_{k=1}^{n-1} \sigma_0(k) \sigma_0(n-k)$$
I ran some quick ...
- 1,949
9
votes
1
answer
558
views
Is the divisor counting function equidistributed mod $p$?
Let $\sigma_0(n)$ be the divisor counting function:
$$\sigma_0(n) = \sum_{d \vert n} 1.$$
I ran some numerical experiments that showed when $p$ is prime, the function $\sigma_0(n)$ is equidistributed ...
- 1,949
4
votes
0
answers
86
views
On Carmichael function and aliquot parts of odd perfect numbers
I've asked nine months ago this question on Mathematics Stack Exchange with identifier 4430381 and same title. There is not answer for this question on Mathematics Stack Exchange, I wondered if this ...
3
votes
0
answers
76
views
Divisor of given order in short intervals
Is the following Open question or Conjecture already known, or eventually settled ?
Open question : For sufficiently large $x$ there is at least a positive integer in the interval $[x,x+\log^2(x)]$ ...
- 433
5
votes
1
answer
187
views
Small covering of divisors
Let $D_n$ be the set of divisors of $n$.
Does there always exists a $B\subseteq D_n$ such that $D_n = \{\gcd(ab,n) \mid a\leq \sqrt{n}, b\in B\}$ and $\sum_{b\in B} \frac{n}{b}=O(n)$?
- 613
4
votes
1
answer
271
views
Around the equation $\sigma\left(\square\right)=\text{prime}$: counterexamples or a proof for some of these conjectures
For integers $A,B\geq 1$ we define the difference $\sigma(A)\sigma(B)-\sigma(AB)$, denoting it as $[A,B]$, where $\sigma(n)=\sum_{1\leq d\mid n}d$ denotes the sum of divisors function. It is possible ...
3
votes
0
answers
179
views
The binary additive divisor problem in arithmetic progressions
I find quite a few results about the binary additive divisor problem, that is evaluating
\[ \sum _{n\leq x}d(n)d(n+h)\]
for certain ranges of $h$.
Are there any known results about the same count ...
- 1,381
1
vote
0
answers
305
views
About inequalities that involve the sum of divisors, the Euler's totient and the aliquot part $\sigma(n)-n$
In this post, for integers $n\geq 1$, I denote the sum of divisors $\sum_{1\leq d\mid n}d$ as $\sigma(n)$ and the Euler's totient function as $\varphi(n)$. It's easy to check* that if we assume that $...
1
vote
0
answers
106
views
Lower bound on a Truncated Divisor Sum
Let $d(n)$ be the number of divisors function, i.e., $d(n)=\sum_{k\mid n} 1$ of the positive integer $n$.
I am interested in estimating, the following sum
$$
A(a,x)=\sum_{n\leq x} \min[ d(n), M]^a
$$
...
- 10.4k
1
vote
1
answer
244
views
Moments of number of interval restricted divisors
I have previously asked the question A truncated divisor function sum
where the sum
$$
S_f(x)=\sum_{n\leq x} \min\{f(x),d(n)\}\quad (1)
$$
was of interest, and it was answered satisfactorily.
Here, I ...
- 10.4k
3
votes
1
answer
436
views
Is there a Riemann Hypothesis criterion utilizing sum of squares of divisors?
Robin's inequality
$$\sigma_1(n)<e^\gamma n\log\log n$$
at integers $n>5040$ provides necessary and sufficient condition for Riemann Hypothesis where $\sigma_1(n)=\sum_{d|n}d$ is sum of divisors ...
- 13.9k
3
votes
0
answers
121
views
On consecutive superabundant numbers
Define $\sigma(n)=\sum_{d\mid n} d$. A number $n>1$ is said to be superabundant (SA) if it is an integer and $\frac{\sigma(n)}{n}>\frac{\sigma(s)}{s}$ for every positive integer $s<n$. Let $n$...
- 31
8
votes
0
answers
271
views
Restricted divisor summatory function
I have a problem that boils down to prove that the succession $\{a_n\}$ tends to infinity, where
$$a_n:=1+\sum_{0\leq j<n}D_{2j+1}(n-j)$$
and $D_{m}(n)$ is the number of divisors $d>1$ of $n$ ...
- 361
7
votes
1
answer
370
views
If $n = 18k+5$ is composite, there are at least 9 divisors of $\phi(n)$ which do not divide $n-1$
If $n$ is a composite of the form $18k+5$, there at least 9 divisors of $\phi(n)$ which do not divide $n-1$. Is this true in general or if not, what is the smallest counter example? The conjecture has ...
- 5,470
2
votes
0
answers
75
views
Least number of factors $\sigma(p^e)$ of representation of $\sigma(N)$ to get the least multiple of $\operatorname{rad}(N)$, for odd perfect numbers
I've cross-posted this from the post of Mathematics Stack Exchange that I've asked (Apr, 2nd 2020) with title On the least number of factors $\sigma(q^{e_q})$ to get the least multiple of $\...
0
votes
0
answers
152
views
On the $\mathsf{LCM}$ of a set of integers defined by moduli of powers
For integers $a,b,t$ define $$\mathcal R_t(a,b)=\{q\in\mathbb Z\cap[1,\min(a^t,b^t)]: a^t\equiv b^t\bmod q\}$$ and $\mathsf{LCM}(\mathcal R_t(a,b))$ to be $\mathsf{LCM}$ of all entries in $\mathcal ...
- 1,826
0
votes
1
answer
146
views
On $\mathsf{LCM}$ of a set of integers
For integers $a,b$ define $$\mathcal R(a,b)=\{q\in\mathbb Z\cap[1,\min(a,b)]: a\equiv b\bmod q\}$$ and $\mathsf{LCM}(\mathcal R(a,b))$ to be $\mathsf{LCM}$ of all entries in $\mathcal R(a,b)$.
How ...
- 1,826
0
votes
1
answer
260
views
Generalized Erdős multiplication table problem
Consider multiplication operation $$f(x_1,\dots, x_k)=\prod_{i=1}^kx_i$$ where $x_i\in\{1,\dots, n_i\}$ with $n_1,\dots, n_k\in\{1,\dots,\infty\}$.
What is the cardinality of the range?
At $k =2$ ...
- 1,826
2
votes
1
answer
198
views
Bounds for two arithmetic functions, when one assumes that $n$ are odd perfect numbers
For an integer $n>1$ in this post we denote the Dedekind psi function as $\psi(n)=n\prod_{\substack{p\mid n\\p\text{ prime}}}\left(1+\frac{1}{p}\right)$ and the product of distinct primes dividing ...
2
votes
0
answers
192
views
The multiplicative constant in the estimate for $S_a(x)=\sum_{n\leq x} d(n)^a$
Let $a$ be a positive real constant and let $d(n)$ denote the number of divisors of $n.$ Define
$$
S_a(x)=\sum_{n\leq x} d(n)^a.
$$
For $a=1,$ the following is well known
$$
S_1(x)=\sum_{n\leq x} d(n)...
- 10.4k
6
votes
0
answers
201
views
Smooth integers with lower bound on $\omega(n)$
Define $(b,c)$-smooth integers to be integers having all prime factors bigger than $c$ and smaller than $b$.
Probability a number is $(b,1)$-smooth is governed by the Dickman function while ...
- 1,826
7
votes
1
answer
231
views
The asymptotic of $|\{1\leq n\leq x|\gcd(n,S(n))=1\}|$, with $S(n)$ the sum of remainders, and get idea for other miscellany problem
Let $n\geq 1$ be an integer. In this post we denote the sum of remainders function as $$S(n)=\sum_{k=1}^n n \bmod k,$$ for example $S(1)=S(2)=0+0$ and $S(5)=0+1+2+1+0=4$. In the literature there are ...
8
votes
0
answers
643
views
Divisor problem: find the fallacy!
The following approach to the divisor problem (that is, the problem of estimating $D(x) = \sum_{n\leq x} d(n)$, where $d(n)$ is the number of divisors of $n$; more precisely, we are meant to bound the ...
- 20.2k
28
votes
3
answers
3k
views
Expressing the Riemann Zeta function in terms of GCD and LCM
Is the following claim true: Let $\zeta(s)$ be the Riemann zeta function. I observed that as for large $n$, as $s$ increased,
$$
\frac{1}{n}\sum_{k = 1}^n\sum_{i = 1}^{k} \bigg(\frac{\gcd(k,i)}{\...
- 5,470
20
votes
2
answers
2k
views
Is every prime the largest prime factor in some prime gap?
Definition: In the gap between any two consecutive odd primes we have one or more composite numbers. One of these composite number will have a prime factor which is greater than that of any other ...
- 5,470
3
votes
1
answer
309
views
How to estimate the sum $\sum_{n\le x} \frac{n}{\tau(n)}$?
Let $\tau(n)$ be the number of positive divisors of $n\in \mathbb{N}$.
Is it possible to get some good estimate for the sum $\sum_{n\le x} \frac{n}{\tau(n)}$?
I know that the sum is $\mathcal O(x^2)$...
- 3,866
0
votes
0
answers
89
views
Partial sums involving Gregory coefficients that cannot be an integer
For integers $n\geq 1$ let $G_n$ be the Gregory coefficients or reciprocal logarithmic numbers, see the Wikipedia [Gregory coefficients.]
(https://en.wikipedia.org/wiki/Gregory_coefficients)
$${z\...
5
votes
0
answers
233
views
What is known about the mode of the number of divisors $\le x$?
Let $d(x)$ be the divisor function. Let $M(x)$ ($x$ a positive integer) be the most frequent value of $d(y)$ for $1 \le y \le x$. Is an asymptotic known for $M(x)$, and failing that, can $M(x)$ at ...
- 251
1
vote
1
answer
258
views
Sum of divisors of Stirling numbers of the second kind
In this post we denote the Stirling number of the second kind as ${n\brace k}$, I add as reference the article Stirling numbers of the second kind from the encyclopedia Wikipedia. And we denote the ...
0
votes
0
answers
47
views
Approximation of $\sum_{\substack{n\geq 1\\n\text{ is abundant}}}\frac{\sigma(n)}{n^3}$, where $\sigma(n)$ denotes the sum of divisors function
Let $\sigma(n)=\sum_{1\leq d\mid n}d$ the sum of divisors function then, from the theory of Dirichlet series, it is well-known the value of $$\sum_{n=1}^\infty\frac{\sigma(n)}{n^3},$$
in terms of ...
5
votes
2
answers
402
views
Estimating $\sum_{n\leq x: n \in A} d(n)^a$ from below for large sets $A\subset \{1,2,\ldots,x\}.$
I apologise for the long-windedness of this question.
Let $a$ be a positive real constant and let $d(n)$ denote the number of divisors of $n.$ Define
$$
S_a(x)=\sum_{n\leq x} d(n)^a.
$$
For $a=1,$ ...
- 10.4k
2
votes
1
answer
280
views
On a problem that equates $\frac{\text{prime}-1}{\operatorname{rad}(\text{prime}-1)}$ with the sequence of primorials
We denote for integers $m>1$ the product of the distinct prime numbers dividing $m$ as $$\operatorname{rad}(m)=\prod_{\substack{p\mid m\\p\text{ prime}}}p,$$
with the definition $\operatorname{rad}(...
11
votes
3
answers
605
views
Number of matrices with bounded products of rows and columns
Fix an integer $d \geq 2$ and for every real number $x$ let $M_d(x)$ be number of $d \times d$ matrices $(a_{ij})$ satisfying: every $a_{ij}$ is a positive integer, the product of every row does not ...
- 213
1
vote
0
answers
222
views
Attempt of exploit the equation $1/\operatorname{rad}(n)=1/2-2\varphi(n)/\sigma(n)$ in the context of even perfect numbers, and a related conjecture
It is well known that the problem concerning even perfect numbers is to prove or refute if there are infinitely many of them. Few weeks ago I wrote the following conjecture, where $\varphi(n)$ denotes ...
1
vote
0
answers
101
views
Size of a set defined by divisor function
After some computations, I guessed the following conjecture.
How can I prove or disprove it? thanks!
Let
$$
A(k)=\#\left\{\left(t,\frac{k+t+a}{4t-1}\right):1\leq t\leq k,\ 1\leq a\leq k+t,\ a\mid(k+...
- 841
2
votes
0
answers
112
views
Queries on distribution of prime divisors by magnitude?
Erdos-Kac law state a typical number of magnitude $n$ has $\log\log n$ prime factors and we know probability of square free integers is $\frac{6}{\pi^2}$.
What is the probability distribution of ...
- 13.9k
3
votes
0
answers
280
views
Magnitude and distribution of largest prime factor?
Erdos-Kac law state a typical number of magnitude $n$ has $\log\log n$ prime factors.
What is magnitude and distribution of largest prime factor of typical magnitude $n$ natural number?
What is ...
- 13.9k
6
votes
1
answer
258
views
How to obtain an upper bound for $\prod_{p\mid N} (1 + 1/\sqrt{p})$ where $N$ is square free?
I am interested in obtaining an upper bound for $\prod_{p|N} (1 + 1/\sqrt{p})$ when $N$ is squarefree. It's not too hard to show that
$$
\prod_{p\mid N} (1 + 1/\sqrt{p}) \ll C^{\omega(N)} \ll N^{\...
- 3,625
4
votes
1
answer
642
views
Piltz Divisor Problem
Let $\tau_k(n)$ count the number of ways of representing $n$ as the product of $k$ natural numbers. It is known that:
$$D_k(x) = \sum_{n \leq x} \tau_k(n) = xP_k(\log x) + O(x ^{1 - \frac{1}{k-1}}(\...
- 151
2
votes
1
answer
283
views
A truncated divisor sum
I am interested in an upper bound for
$$\sum_{\substack{d\mid N\\ d>A}}\frac{1}{d^3},$$
in particular, I can show that above is
$$\ll\frac{\text{exp}\left(C\frac{\log(N)}{\log\log(N)}\right)}{A^...
- 23
8
votes
1
answer
1k
views
Sum of divisors below threshold
Let $\sigma(n)$ denote the sum of divisors of $n$, that is,
$$
\sigma(n) = \sum_{d | n} d.
$$
It is known that $\sigma$ can have values as large as order $n \log \log n$. However, obviously the sum is ...
- 2,207
1
vote
0
answers
141
views
On certain number theoretic sextuples?
Given small parameters $0<\epsilon<\epsilon'$ is there an $n_\epsilon>0$ such that at every $n>n_\epsilon$ if we are given a prime $n^2<p<2n^2$ then can we always find integers $a,b,...
- 13.9k
5
votes
3
answers
716
views
Good books on the divisor sum function $\sigma(n)$?
I would like gain detailed knowledge about properties of the divisor sum function $\sigma(n)$, special equation that have been studied (e.g. $\sigma(n) = 2n$ perfect numbers, ...) and progress that ...
- 159
7
votes
1
answer
675
views
Short divisor sum
Let $d(n)$ denote the number of positive divisors of the positive integer $n$.
Pick some positive $X,h \in \mathbb{R}$ and consider the sum
$$ S(X,h) := \sum_{X \leq n \leq X+ h} d(n).$$
In view of ...
- 11.3k
1
vote
1
answer
233
views
Generalized notion of divisor function?
Divisor function $d(n,m)$ counts the number of $q\in\Bbb N$ with $b<q<m$ such that $n\bmod q\equiv0$.
Given $b>0$ what is the correct asymptotic, probabilistic and average case behavior of ...
- 13.9k
1
vote
0
answers
90
views
An arithmetic function involving arbitrary (but fixed) number of divisors
I need at least basic information about generating functions of the following class of arithmetic functions, grouped by levels $k$.
Fix some $k$ and some family $\varepsilon_*=(\varepsilon_\sigma)_{\...
- 18.4k